[/quote]
,
with incenter 
,
touches sides 
and 
at points 
and 
,
and 
.
meets 
at 
.
is parallel to 
.
with incircle 
be the point on 
-
.
where 
and 
Prove that
Let 
Prove that
Let 
Prove that
on segment 
Find the ratio 
,
is a point on arc 
is maximized.
such that for all real numbers 
,
be distinct real numbers such that 
and 
Evaluate all the possible values of 
.
.
be an acute scalene triangle with orthocenter 
.
intersects 
at 
and line 
intersects 
at 
.
be the foot of the perpendicular from 
parallel to 
.
lies on line 
such that 
is parallel to 
lies on line 
such that 
is parallel to 
,
,
satisfy that 
.![$$\sum_{cyc} \sqrt[4]{\frac{a^3}{b+c}}$$](http://latex.artofproblemsolving.com/e/f/0/ef089c4f2915bbf75b0f60b044bbe9273b301a81.png)
=2$$](http://latex.artofproblemsolving.com/c/8/b/c8b33f21c28cd9d193abb2848116835353aeb8bd.png)
. The incircle 
is tangent to 
at 
respectively. A line through 
,
intersects 
.
is the midpoint of 
be the second intersection of 
with 
intersects 
bisects 
.
Y by
Round 2
1 Define the isolation index of real number
to be 
, where ![$[x]$](//latex.artofproblemsolving.com/b/c/e/bceb7b14e55d33a8bca29b7863ad3cdae95afce4.png)
is the largest integer no greater than 
, and ![$\{x\}=x-[x]$](//latex.artofproblemsolving.com/6/1/0/61036a03415081af535bd57c408b7be79aae9458.png)
. For each positive integer 
, find the maximum possible real number 
such as there exists an infinite geometric sequence with common ratio and isolation index of each term being at least .
1 Define the isolation index of real number
to be 
, where ![$[x]$](http://latex.artofproblemsolving.com/b/c/e/bceb7b14e55d33a8bca29b7863ad3cdae95afce4.png)
is the largest integer no greater than 
, and ![$\{x\}=x-[x]$](http://latex.artofproblemsolving.com/6/1/0/61036a03415081af535bd57c408b7be79aae9458.png)
. For each positive integer 
, find the maximum possible real number 
such as there exists an infinite geometric sequence with common ratio and isolation index of each term being at least .This post has been edited 2 times. Last edited by scls140511, Sep 8, 2024, 2:24 PM
.
